March 9, 2022

Spatial-structure

Simple population: \[\large N(t)\]

Age-structured: \[\large N_i(t) = \{N_1(t), N_2(t), ..., N_k(t)\}\]

  • where \(i\) represents age structure, with \(k\) age stages

Spatial structure: \[\large N_i(t) = \{N_1(t), N_2(t), ..., N_k(t)\}\]

  • where \(i\) is location, with \(k\) locations

A metapopulation …

… is a population of populations

The local populations MUST be somehow connected via dispersal.

There must be areas of (near) zero density in between. The “in-between” is referred to as the matrix.

Examples

  • Island populations
  • Fragmented habitats
  • Humans
  • Just about anything with genetic structure!

“Population” vs. “Metapopulation”

WA sea otters

  • closed population
  • individuals living in same place, same time
  • only Birth and Death
  • Interested in: growth / dynamics / age structures
  • Extinction mainly relevant under stochastic scenarios

All sea otters

  • open population
  • individuals in different places, at the same time
  • Immigration and Emigration key processes
  • the main question of interest is presence / absence
  • given that local populations go extinct, will the metapopulation go extinct?
  • what is the proportion of occupied patches?

Population persistence

\(e\) = local probability of extinction

  • 1 time step: \(1-e\)
  • 2 time steps: \((1-e)(1-e)\)
  • 3 time steps: \((1-e)(1-e)(1-e)\)
  • 4 time steps: \((1-e)(1-e)(1-e)(1-e)\)
  • \(t\) time steps: \((1-e)^t\)

Take away: Even with very LOW probability of extinction, you WILL go extinct.

Metapopulation persistence

\(k\) populations, \(t\) time steps

Pops: One time step t steps
1: \(1-e\) \((1-e)^t\)
2: \(1-e \times e\) \((1-e^2)^t\)
3: \(1-e \times e \times e\) \((1-e^3)^t\)
k: \(1-e^k\) \((1-e^k)^t\)

Metapopulations are resistant to extinction

Metapopulations DRAMATICALLY spread out the risk of extinction!

But still - if ONLY process is extinction … you WILL go extinct (sorry!)

\[P(k,t) = (1-e^k)^t\]

M1: Let’s add colonization

Island-Mainland model

  • Every (local) population has a probability of going extinct: \(p_e\)
  • But every empty location has a probability of getting colonized: \(p_c\)

Note - there is also here an important (implicit) assumption that population very quickly hits carrying capacity, so essentially instant saturation.

The mainland is a constant, independent source of potential colonizers. Also known as propagule rain.

(echoes of biogeography).

M1: Island-Mainland Model

Q: How many occupied patches might we expect?

\[E(N_{t+1}) = N_t - p_e N_t + (K - N_t)\,p_c\]

define proportion of populated patches: \(f_t = E(N_t)/K\), and define equilibirum: \[f^* = f_{t+1} = f_t\]

…then some math happens…

\[\large f^* = {p_c \over p_c + p_e}\]

Continuous time formulation

Very general metapopulation model: \[\large {df \over dt} = c - e\]

Where \(c\) - colonization rate, \(e\) - extinction rate.

Note: this is similar to \({dN \over dt} = b - d\).

(Highly unrealistic) assumptions!

  • Deterministic, i.e. number of populations \(K \to \infty\)
  • Continuous, unstructured extinction / colonization process
  • “Rates” are like infinitesemal probabilities

But - lots of elegant conclusions can be made messing with this model.

M1: Mainland-Island

\[\large {df \over dt} = c - e\]

Colonization is constant, so proportional to available patches: \[c = p_c(1-f)\]

Extinction is constant, so proportional to occupied patches:

\[e = p_e f\]

so: \[ {df \over dt} = p_c(1-f) - p_ef\]

in words: The rate of change of the occupied patches GROWS in proportion to unoccupied patches and FALLS in proportion with occupied patches.

M2: Internal Colonization

\[ {df \over dt} = p_c f(1-f) - p_ef\]

Extinction is constant, as before: \[e = p_e f\]

Colonization can only come from occupied patches: \[c = p_c \, f \, (1-f)\]

If no patch is colonized (\(f=0\)), nothing can colonize.

If the population is 100% occupied (\(f = 1\)), there is nothing to colonize.

M2: Internal Colonization - with Schematic

\[ \large {df \over dt} = p_c f(1-f) - p_ef\]

  • If no site is colonized (\(f=0\)), nothing can colonize.
  • If the population is 100% occupied \((f = 1)\), there is nothing to colonize.
  • The maximum rate of colonization occurs when \(f = 1/2\).
  • Equilibrium occurs when: \[f^* = \begin{cases} 1-p_e/p_c & \text{when} & p_e < p_c \\ 0 & \text{when} & p_e \geq p_c\end{cases}\]

M3: Rescue Effect

\[ \large {df \over dt} = p_c (1-f) - p_ef(1-f)\]

Assumes that if you have a lto of neighbors some loose “propagules” will buffer you from extinction.

Equilibrium states:

\[f^* = \begin{cases} p_c/p_e & \text{when} & p_e > p_c \\ 1 & \text{when} & p_e \leq p_c\end{cases}\]

Even with higher extinction rate than colonization rate, there will always be some occupied patches!

M4: Rescue Efffect with Internal Colonization

\[ \large {df \over dt} = p_c f(1-f) - p_ef(1-f)\]

Only equilibria: 0, if \(p_e > p_c\) or 1, if \(p_e \leq p_c\).

Fundamental conclusions: metapopulation under equilibrium MUST be rare! - either everything colonizes or nothing colonizes.

Four models

With rather different predictions! (Nice synthesis - mainly due to Gotelli.)

Some characters

Richard Levins (1930-2016)

  • “Scholarship that is indifferent to human suffering is immoral.”
  • “Our truth is the intersection of independent lies.”

Ilkka Hanski (1953-2016)

Nice theory you’ve got here ….

but how can it be applied with so many unrealistic assumptions?

  • deterministic processes
  • “infinite” number of patches
  • homegeneous patches
  • homogeneous process
  • “instantaneous” population growth
  • implicit spatial structure - (all patches affect all others equally)

With case-specific modeling!

Which is it!?

It’s hard to do metapopulation studies!

Example 1: Abalone intro

Pinto / northern abalone (Haliotis Kamtschatkana) - overharvested commercially in British Columbia, commercial harvest banned, recovering.

Question: how is “poaching” affecting their recovery?

Example 1: Abalone model

If you look closely…

  • Growth rate \(r\)
  • Carrying capacity \(K\)
  • Age structured fecundity
  • Survival
  • Dispersal distances
  • Dispersal survival

Example 1: Abalone results

Used: metapopulation probability of extinction = 0.1 as threshold, corresponding to IUCN definition of Vulnerable.

Metapopulation management

  1. Can be challenging because equilibrium might not exist!
  2. Metapopulation will surely become extinct if patches are removed …
  3. … but facilitating recolonization and maintaining large patches can help.
  4. As many fragments as possible should be preserved…
  5. … but distances can’t be too large, or no recolonization or rescue effect.
  6. Properties of the matrix are important: corridors and stepping stones.
  7. Recolonization has to be observed within a few generations for metapopulations to have a chance.
  8. Sizes of patches is important to hedge against demographic stochasticity.

(from: Hanski, I. 1997. Metapopulation biology. Pp. 69-91. San Diego, USA, Academic Press.)